Reflections on maths pedagogy, education, and teaching craft

This is the second post in a series. In Part 1, I reflected on the vast numbers of 16 year olds who finish 11 years of maths education unable to answer the straightforward questions needed for a grade C. I argued that this is and continues to be a problem, despite rising C-grade pass rates over the past 20 years. This post reflects more deeply on the causes behind why so many 16 year olds are in this state.

Matthew Syed’s ‘Black Box Thinking’ is a great read. He examines extraordinary successes in various fields like medicine, healthcare, aviation safety – like Team Sky, or Google – to find what they have in common. The answer? Such successes ‘harness the power of failure’. They don’t shy away from mistakes, but rather expect them, learn from them, and milk them for all the information they are worth. The aviation industry does this extremely well:

Pilots are generally open and honest about their own mistakes (crash-landings, near misses). The industry has powerful, independent bodies designed to investigate crashes. Failure is not regarded as an indictment of the specific pilot who messes up, but a precious learning opportunity for all pilots, all airlines and all regulators.

Some examples from Syed that bear this culture out: flight accidents are automatically transferred to independent investigators to look into, and those involved in the flight are protected to full disclosure, as whatever they say is inadmissible in court. The report is then circulated and freely available to any pilot in the entire world, ensuring that the entire industry can learn from the accident. (This is a striking comparison to the education sector in this country– but that would require another blogpost in itself…)

As Syed goes on to explain, this culture is particularly useful since the dissemination and circulation of information on failures can quickly reveal ‘signatures’ – particular patterns that keep recurring in various mistakes, problems, and accidents. For example, in one week in 2005, a whole host of reports of near-misses came from pilots landing in Lexington Airport. The investigators quickly cottoned onto the problem from the info from the pilots’ reports: lights had just been installed on an adjacent plot of land, which was then being mistaken for the runway. Within days (apparently, quite slow for the industry – the adjacent land didn’t belong to the airport) the confusing lights were taken down.

In the spirit of this ‘black box thinking’, I’ve thought about what problems and mistakes I frequently observe in my own classroom, together with the mistakes and honest frustrations I hear from colleagues across various schools. I’ve focused particularly on my experience with bottom sets, since it is these groups that usually fail to make grade C (though my conclusions may be more widely true too). From the mistakes with these classes, what are the ‘signatures’ – the common recurring patterns? What are the issues I keep experiencing and hearing about from other teachers? And, in the following post – what way forward can be discerned from these signatures, from these diagnoses?

In 2014, 62.4% of teenagers sitting GCSE maths achieved at least a grade C – the ‘gold standard’ grade needed to progress onto a wide range of further study and employment. For the last 5 years, the pass rate has hovered around 60%.

The flipside of this statistic: every year, at least 40% – that’s a quarter of a million 16-year olds – complete their secondary education without having attained a C in GCSE maths.

Is this the best we can do?

I looked at the 2014 GCSE Higher maths papers. To have gotten a grade C in it, you needed 57 marks out of 200 (a statistic that shocked me when I first discovered it, and continues to shock). Below are 63 marks’ worth of questions from 2014’s two papers (click to enlarge) – if you got all of these questions right, give or take 2-3 of them, you would’ve done enough to get that coveted grade C.

To repeat: in 2014, as in other years, at least a quarter of a million 16-year olds finished 11 years of compulsory education unable answer all these questions.

Is this the best we can do? I hope not. Here’s some preliminary reflections and thoughts on this state of affairs.

‘Why on earth are you a teacher? I still don’t understand. You could’ve done so many things and you chose to become a teacher.’

– Multiple pupils

Respect in a profession is not the most important thing. Take rubbish collectors and lawyers, for example. Many jobs are not particularly respected – they may even be looked down upon – yet they nonetheless play an important part in society. They may even be relatively well compensated.

Nonetheless, respect is important. A respected profession is one which attracts high quality applicants. Given the teacher recruitment problems in the UK, it’s worth thinking about respect for the profession.

It’s also worth thinking about respect for the sake of well-being. Teachers are demotivated. While there are a huge variety of causes for this, a sense of general respect in teaching would undoubtedly help.

I attended the Michaela debate on Saturday 23rd April. The debate on ‘No excuses discipline works’ had me thinking the most, with the barrister-esque Jonathan Porter speaking against the wise and fatherly John Tomsett. It was a brilliant debate. They have both since posted their transcripts online, and other prominent bloggers who were present have offered their responses; this has in turn produced a lotofreaction from people not present for the debates.

All this has shown sharp divisions in the first order debate: people’s opinions and interpretations of ‘no excuses’ discipline and what it means. Yet it’s clear from the sharp divide that there are serious second order issues: what a behaviour system is for, and what values ought to underpin it.

It’s increasingly clear that big ethical questions pervade and underlie this entire discussion. For example, both sides grapple with the idea of fairness:

So, you’re convinced of the worked example effect, and you use it in lessons with great regularity and regular success. Yet have you ever worked through an example only for your students to be completely flabbergasted?

This happened to me the first time I taught quadratic simultaneous equations to my year 11s (set 2/4). They were pretty solid in solving quadratics and at simultaneous equations by elimination – we’d just spent half the lesson revising them. I put this example up on the screen:

and I duly worked through it, rearranging 2, substituting into 1, reminding them of how to expand a binomial squared, expanding and simplifying and solving and then finally substituting again. In fairness I should have stopped much earlier, from the puzzled sounds that were emanating from the class. But it soon became clear from questioning that students had absolutely no idea what had just happened. The problem: total cognitive overload. Whoops! I tried to start again, emphasising each step, but I had lost most of the class’s motivation, and we duly went back to revising quadratics and simultaneous equations.

Fast forward a term (I hoped that was enough time for them to recover from my first attempt). Same class, same topic – yet very different approach. I didn’t start with the problem, or work through an example. Instead, I tackled through the problem in reverse. Again, the starter was a mix of quadratic equations of varying complexity. But after we’d revised that, this was my sequence of examples, in this order:

Slide 1

This was the first slide – and it was very easy compared to lots of the questions in our starter. I picked a student to solve it for the class, and she quickly did.

Slide 2

This was slightly more complicated, but again within the reach of the class. I again picked a student, asking what was different, and then asking them to solve it. However, upon rearranging, many students immediately recognised that it led to the same problem as before, and so we already knew the solutions.

Slide 3

Slide 3 – a great way to highlight some misconceptions – I picked a mid-attaining student from the class who correctly identified the first step as expand the brackets, but incorrectly stated it as the classic ‘x squared plus 4’. Students discussed this briefly (whilst I went mock-apoplectic at him), and thankfully other students quickly corrected him. After that pause, we simplified and saw – it was exactly the same as the problems in the previous slides, and again we knew the answers. And then finally:

Slide 4

Quadratic simultaneous equations – finally, our new bit of learning. I introduced the concept of expression substitution to them, and showed how it worked, and the resulting equation that formed. And once again, to no-one’s surprise by this stage, we were led to exactly the same equation that we had had in the previous slide. As a result, the rest of the process was trivial – I simply told them to ‘solve the remaining equation as we just did’, flicked back each slide (where we’d solved each problem on the board), and students knew exactly what I meant. I then immediately went back to the first slide, and the solution we’d arrived at there, and asked if that was the final answer. Impressively, a large number of students were able to immediately identify that we had to solve for y as well. It was clear that this method of explaining in reverse had avoided the catastrophic working memory overload of my first attempt. And, when it came to trying problems, the entire class (ranging from predicted grades of Cs to A*s) were able to solve at least several problems successfully.

These two experiences couldn’t have been more different. The only difference was the un-intuitive – but highly successful – reverse ordering of my explanation. Like many good ideas, I originally encountered this from Kris Boulton, who used it in the context of linear simultaneous equations; I’m blogging about it just because I haven’t seen it mentioned anywhere else, but it’s been massively helpful to me in this instance and several others, and I’ve wished I’d used it more in other topics.

I think the key pedagogical insights of this method is as follows:

In worked examples, focus attention and cognitive load, as far as possible, on the new concepts/processes/facts that you want pupils to learn.

Recognise that if a worked example contains parts that pupils are already familiar with, it nonetheless contributes to cognitive load, and thus may hinder their ability to grasp what’s new.

Therefore, if a worked example involves steps that utilise lots of other previously-learnt things, work through the problem in reverse, establishing the most basic steps before you move onto what’s new.

In the example above, the new concept was of substituting between equations, and the rest of the example reduced to things that pupils already knew how to do. Yet as my first experience showed, those parts of the example involving rearrangement/simplifying/solving quadratics overloaded my pupils, who were undoubtedly still trying to get to grips with the idea of equation substitution. As my second experience showed, far better to start with what they felt comfortable with, establish it (and in the case of (x+2)^2, get the peripheal & familiar misconceptions out of the way there), and work backwards until we reached what was new.

This method can be used in pretty much any topic which involves multiple steps. It can work in KS3 – for example, when teaching ordering fractions, decimals and percentages:

First example/problem: Convert 1/2, 1/4, 3/10 into decimals

Second problem: ‘Order 0.6, 0.2, 0.5, 0.25, 0.62 in ascending order.’

Worked example: ‘Order 3/10, 0.2, 1/2, 1/4, 0.62 in ascending order.’

A similar principle also works in KS5 teaching too, especially well, as most questions are rather long. Since questions here have numerous parts and it’s a bit harder to separate them out, I often apply a slightly different but related approach: pre-provide everything that the students already know how to do in the example, so that they can focus on the new content.

For example, the Edexcel C2 textbook offers something like this example for binomial expansion approximations: ‘Find the first four terms of (1+8x)^6, and by using a suitable substitution, use your result to find an approximate value to (1.016)^6.’ Yet the only thing that’s new is the last bit – approximations using substitutions. So, when I explained this new process, I simplified the example by providing the familiar part of it for them: ‘The first four terms of (1+8x)^6 are 1 + 48x + 960 x^2 + 10240 x^3. By using these and a suitable substitution, use your result to find an approximate value to (1.016)^6′.

By pre-providing the first part of the answer within the example, we instantly honed in on the new content, directed all our attention to that, had lots of time to ask questions about it, and weren’t held up by having to perform the expansion. Interestingly, when students eventually did see an exam-style question, they twigged by themselves the need to expand it first, and happily did so. Similarly, with my year 11s and the simultaneous quadratics, most of the class worked out for later questions (with a bit of discussion with their partner) that they had to rearrange some equations before substituting them. I think this happened because, having been able to focus entirely on the new process during the reverse explanation, the students had already grown quite comfortable with it, and were thus able to apply other parts of their prior knowledge to new and harder questions without much further instruction from me.

We naturally want to introduce what’s new first. Yet if this is then followed by lots of other familiar content in order to get to the final answer, we run the risk of cognitive overload in our pupils. As teachers who have mastered the curriculum, this often doesn’t occur to us, so I’m thankful for that original lesson with my year 11s for bringing this strategy back to mind. Either reverse your explanations so more familiar final steps are addressed first, or just pre-provide the familiar steps of an example, so that you can focus pupils’ attention and working memory on what’s new.

We felt very nice and snug, the more so since it was so chilly out of doors; indeed out of bed-clothes too, seeing that there was no fire in the room. The more so, I say, because truly to enjoy bodily warmth, some small part of you must be cold, for there is no quality in this world that is not what it is merely by contrast. Nothing exists in itself.

– Herman Melville, Moby Dick

Variation theory: you can’t fully understand a concept, unless you understand what it is not.There are loads of brilliant applications of this in teaching. Kris Boulton first introduced the concept to me via GCSE students who were unable to identify this shape as a pentagon:

The reason? While they might have been told that a pentagon is a shape with 5 sides, what they usually see accompanying this explanation is:

and it’s this regular pentagon that sticks in their minds as a ‘pentagon’, leaving them at a loss when faced with naming the unfamiliar shape above.

So, when defining the concept of a pentagon, it’s most helpful to show numerous examples of various irregular pentagons, together with shapes which are not pentagons, so learners can develop a full concept of what ‘having 5 sides’ means and doesn’t mean. It’s another way of addressing the problem Greg Ashman highlights here:

To experts, it’s obvious that the shape above is a pentagon, due to a thorough understanding of pentagons through many experiences; yet to communicate only that “a pentagon is ‘a shape with 5 sides’ that looks like ⬠” means that many learners will link the concept only with what’s been highlighted – the regular pentagon.

In this post I’ll look at one easy way to apply variation theory in designing tasks: adding peripheral questions to tasks.

I worry sometimes about doing differentiation for its own sake. We don’t talk so much about the goals or the assumptions behind differentiation, when those, to me, seem to be the much bigger questions. In this post I’ll explore some of those questions, and present my view of the conditions when, and how, differentiation makes sense. (The title gives you a hint.)

I remember watching this episode from the Simpsons on BBC2 when I was at school myself. This particular section has lingered in my mind since then, and since becoming a teacher it naturally came back to mind. As I watched it again, it’s even more devastating than I remembered.

While that clip has so much worth commenting on, Bart’s quote says it all:

“Let me get this straight. We’re behind the rest of our class and we’re going to catch up to them by going slower than they are?”